X-rays are a form of electromagnetic radiation with wavelengths in the region of about 0.01 to 10 Angstroms (Å). These wavelengths are short compared to that of visible light, which has wavelengths in the range of about 4000 Å to 7000 Å. Visible light can be easily controlled and focused using known reflective and refractive optics. Visible light optic components do not work with x-rays because x-ray wavelengths approach the dimensions of distances between atoms in solids.
It is known that x-rays are diffraction scattered in all directions by single atoms. If the atoms are arranged substantially in multiple parallel planes (such as the planes in a crystal), there is an opportunity for the scattered x-rays from different planes to interfere constructively with one another. Scattered rays which obey Bragg's Law (described below) will appear to be reflected from the planes. This phenomenon is widely used in the study of crystal structures by x-ray diffraction.
It is also possible to diffract (reflect) x-rays using thin films of alternating layers of high-Z material and low-Z material, where Z is the atomic number of the metal and where the spacing between the layers is on the order of Angstroms. The nanolayers which form such films are analogous to the planes in a perfect single crystal, and scatter x-rays according to the same laws of physics. The effect of diffraction (reflection) from a such a multilayer coating is shown below in FIG. 1.
In FIG. 1. an incident beam 102 of x-rays impinges at an incident angle 112 of magnitude Θ upon a multilayer coating that includes layers 104, 106, 108 and 110 which are uniformly spaced a distance d apart from each other, respectively. Some of the x-rays of beam 102, representing a beam 116 of lesser intensity than beam 102, are diffracted (reflected) at a scattering angle 114 of the same magnitude, Θ. Some of x-rays of beam 102 pass through layer 104 as beam 102′ to layer 106. Upon reaching layer 106, some the x-rays of beam 102′ are diffracted (reflected) as a lower intensity beam 116′ again at the same scattering angle Θ, while some pass through layer 106 as beam 102″ to layer 108. Upon reaching layer 108, some the x-rays of beam 102″ are diffracted (reflected) as a lower intensity beam 116″ at the scattering angle Θ, while some pass through layer 108 as beam 102′″ to layer 110. Similarly, at least some of the x-rays of beam 102′″ are diffracted (reflected) by layer 110 at the scattering angle Θ as a lower intensity beam 116′″, while some (not depicted) may pass through layer 110 and subsequent layers (if present).
Bragg's law describes the condition of diffraction (reflection) depicted in FIG. 2. Bragg's law is as follows.nλ=2d sinΘ  (1)Here, again, λ is the x-ray wavelength, d is the spacing between the layers, Θ is the incident angle (also described as the grazing angle) and n is any non-zero positive integer representing the number of pairs of respective layers of high-Z and low-Z material . By carefully choosing the spacing, d, and the incident angle, Θ, the wavelength, λ, of the diffracted (reflected) light can be controlled to produce specific narrow bandwidths of x-rays.
Bragg's law is applied in the copending '927 application, where a sheaf 246 of stacked rectangular reflectors can act as a filter to produce a narrow band of x-rays, as is depicted in Background Art FIG. 2 according to the copending '927 application. There, a broad band beam 257 of x-rays is depicted as originating from a source 256 and impinging upon a front end of simplified sheaf 246 of reflectors. While sheaf 246 includes a total of 1, 2, . . . , N reflectors, its depiction is simplified, e.g., in the sense that only reflectors 232-N, 232-N-1 and 232-N-2 are depicted. Another simplification, e.g., is that no structures that establish relative spacing between adjacent reflectors 232-i and 232-i-1 are depicted. Further simplifications in FIG. 2 are that relative proportions, e.g., between distances lfi, lli, df1 & dri and ith thicknesses of the reflectors, respectively, and angles α1i and α2i, respectively, are not to scale.
In FIG. 2, an ith distance, di, between front ends of any two adjacent reflectors is substantially the same, i.e., the distance df1 between front ends of reflectors 232-N & 232-N-1 substantially equals the distance df2 between front ends of reflectors 232-N-1 & 232-N-2, etc., namely df1≈df2 . . . . To ensure that each reflector is oriented so that the front end thereof experiences substantially the same incident angle of x-rays, adjacent reflectors are rotated relative to one another. More particularly, to ensure that α1′≈α1, reflector 232-N-1 is rotated a non-zero angle β2 relative to, e.g., horizontal, where it is assumed in FIG. 2 that reflector 232-N is oriented to be horizontal, i.e., its angle, β1, is zero (β1=0). Similarly, so that α1″=α1′, reflector 232-N-2 is rotated an angle β3, where β3>β2, etc. Thus, in FIG. 2, the following is true.α1≈α1′≈α1″  (2)Despite such relative rotation, however, distance, dri, between the rear ends of adjacent reflectors 232-i & 232-i-1 is substantially the same, i.e., a distance dr1 between reflectors 232-N & 232-N-1 is substantially the same as a distance dr2 between adjacent reflectors 232-N-1 & 232-N-2, etc., namely dr1 dr2 . . . .
As a consequence of such relative rotation, the distance dfi between front ends of adjacent reflectors 232-i & 232-i-1 is significantly smaller than the distance dri between rear ends of adjacent reflectors 232-i & 232-i-1, which can be restated as follows.dfi<dri  (3)
Returning to Bragg's law, it describes the wavelength, λi, diffracted (reflected) at an ith point along a reflecting side 233-i of each reflector 232-i. Thus, a wavelength diffracted (reflected) at the front end of reflector 232-i, namely wavelength λf, would be as follows.
                              λ          f                =                              2            ⁢                          t              fi                        ⁢            sin            ⁢                                                  ⁢            α1            ⁢                                                  ⁢            i                    n                                    (        4        )            where tfi is a thickness of the reflecting layers at the front end of reflector 232-i, α1i is the incidence angle at the front end of reflector 232-i, and n is the number of reflecting layer interfaces in reflector 232-i. Similarly, a wavelength diffracted (reflected) at the rear end of reflector 232-i, namely wavelength λr, would be as follows.
                              λ          r                =                              2            ⁢            t            ⁢                                                  ⁢                          i              r                        ⁢            sin            ⁢                                                  ⁢            α2            ⁢                                                  ⁢            i                    n                                    (        5        )            where tri is a thickness of the reflecting layers at the rear end of reflector 232-i, α2i is the incidence angle at the rear end of reflector 232-i, and n (again) is the number of reflecting layer interfaces in reflector 232-i.